Optimal. Leaf size=88 \[ -\frac{2045 \sqrt{1-2 x}}{2058 (3 x+2)}-\frac{545 \sqrt{1-2 x}}{147 (3 x+2)^2}+\frac{121}{14 \sqrt{1-2 x} (3 x+2)^2}-\frac{2045 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}} \]
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Rubi [A] time = 0.0229986, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {89, 78, 51, 63, 206} \[ -\frac{2045 \sqrt{1-2 x}}{2058 (3 x+2)}-\frac{545 \sqrt{1-2 x}}{147 (3 x+2)^2}+\frac{121}{14 \sqrt{1-2 x} (3 x+2)^2}-\frac{2045 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}} \]
Antiderivative was successfully verified.
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Rule 89
Rule 78
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(3+5 x)^2}{(1-2 x)^{3/2} (2+3 x)^3} \, dx &=\frac{121}{14 \sqrt{1-2 x} (2+3 x)^2}-\frac{1}{14} \int \frac{-610+175 x}{\sqrt{1-2 x} (2+3 x)^3} \, dx\\ &=\frac{121}{14 \sqrt{1-2 x} (2+3 x)^2}-\frac{545 \sqrt{1-2 x}}{147 (2+3 x)^2}+\frac{2045}{294} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=\frac{121}{14 \sqrt{1-2 x} (2+3 x)^2}-\frac{545 \sqrt{1-2 x}}{147 (2+3 x)^2}-\frac{2045 \sqrt{1-2 x}}{2058 (2+3 x)}+\frac{2045 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{2058}\\ &=\frac{121}{14 \sqrt{1-2 x} (2+3 x)^2}-\frac{545 \sqrt{1-2 x}}{147 (2+3 x)^2}-\frac{2045 \sqrt{1-2 x}}{2058 (2+3 x)}-\frac{2045 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{2058}\\ &=\frac{121}{14 \sqrt{1-2 x} (2+3 x)^2}-\frac{545 \sqrt{1-2 x}}{147 (2+3 x)^2}-\frac{2045 \sqrt{1-2 x}}{2058 (2+3 x)}-\frac{2045 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}}\\ \end{align*}
Mathematica [A] time = 0.0381378, size = 69, normalized size = 0.78 \[ \frac{21 \left (12270 x^2+17305 x+6067\right )-4090 \sqrt{21-42 x} (3 x+2)^2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{43218 \sqrt{1-2 x} (3 x+2)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 57, normalized size = 0.7 \begin{align*}{\frac{18}{343\, \left ( -6\,x-4 \right ) ^{2}} \left ( -{\frac{133}{18} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{917}{54}\sqrt{1-2\,x}} \right ) }-{\frac{2045\,\sqrt{21}}{21609}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{242}{343}{\frac{1}{\sqrt{1-2\,x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.77345, size = 112, normalized size = 1.27 \begin{align*} \frac{2045}{43218} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{6135 \,{\left (2 \, x - 1\right )}^{2} + 59150 \, x + 5999}{1029 \,{\left (9 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 42 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 49 \, \sqrt{-2 \, x + 1}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63469, size = 244, normalized size = 2.77 \begin{align*} \frac{2045 \, \sqrt{21}{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \,{\left (12270 \, x^{2} + 17305 \, x + 6067\right )} \sqrt{-2 \, x + 1}}{43218 \,{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.14078, size = 104, normalized size = 1.18 \begin{align*} \frac{2045}{43218} \, \sqrt{21} \log \left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{242}{343 \, \sqrt{-2 \, x + 1}} - \frac{57 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 131 \, \sqrt{-2 \, x + 1}}{588 \,{\left (3 \, x + 2\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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